Crystallography. Duration: three weeks (8 to 9 hours) One problem set. Outline: 1- Bravais Lattice and Primitive

Transcription

1 Crystallography Duration: three weeks (8 to 9 hours) One problem set Outline: 1- Bravais Lattice and Primitive 2- Cubic lattices, Simple, Body-centered and Face-centered 3- Primitive Unit Cell, Wigner-Seitz Cell and Conventional Cell 4-Lattices with a Basis: Crystal structures 5- Common crystal structures, hexagonal close packed diamond, sodium chloride and cesium chloride

2 Point Space: The set consisting of elements that are points. It is a model used to describe objects without specifying the exact nature of what we represent by points. Vectors are used to specify a point. Ideal Crystal: An infinite repetition of identical groups of atoms. Lattices: the mathematical points to which the repetitive pattern of atoms is attached is called a Lattice (3D) or Net (2D)

3 Example:of a Net of Lattices!! However!! This is ONLY an aproximation: Neither the 2D array, nor the 3D array are infinite!! Magnetic Cobalt nanoparticles

4 Bravais Lattices Definition 1: A Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same from any point in the array

5 Bravais Lattices (Another) Definition 2: A Bravais 3D-lattice is the sub-set of point space consisting of all points with position vectors given by: R= na 1 1+ n2a2+ n3a3 Where a, a, a are three Vectors not all in the same plane, and n 1, n 2,n 3 are any integer a, a, a The vectors are called primitive vectors and they are said to generate the lattice

6 Simple cubic lattice Are definitions 1 and 2 satisfied? a 1 a 3 a 2

7 Does this net satisfy definitions 1 and 2?

8 The choice if the primitive vectors is not unique. Do all these vectors satisfy definition 2? Vectors a 1 and a 2 are not primitive vectors because we can not reach all points according to definition 2 using integer coefficients

9 Some Important Examples: The Body-centered cubic lattice (bcc) Do all points have equivalent environment? Points B can be seen as corner points or as center points or as body center points.

10 The Body-centered cubic lattice (bcc) One set of primitive vectors

11 The Body-centered cubic lattice (bcc) A more symmetric set of primitive vectors

12 Some Important Examples: The face-centered cubic lattice (fcc)

13 The face-centered cubic lattice (fcc): a set of symmetricprimitive vectors

14 Summary of Bravais Lattices: A periodic array of points is called a Bravais lattice (3D) or Bravais net (2D) if definitions 1 and 2 are satisfied. A set of vectors is always associated with any Bravais lattice (net) These vectors are called primitive and: 1) Are not unique 2) Are Independent of the choice of the origin 3) Generate or span an infinite lattice (net) Physically, the primitive vectors represent a displacement or translation of a repetitive unit. Thus a Bravais lattice (net) is an array of points that is left invariant after any translation through the primitive lattice vectors. Translational Symmetry

15 Some more definitions: Near neighbors and Coordination numbers: The points in a Bravais Lattice that are closest to a point than to any other point are called near neighbors. Each point of a Bravais lattice has the same number of near neighbors This is a property of the lattice The number of near neighbors is called coordination number of the lattice

16 Properties primitive unit cell: 1) It contains only one point. Primitive Unit cell 2) It has a volume that is independent of the choice of primitive cell 3) For any pair of different primitive cells, one can always sub-divide one of them, and by translation through lattice vectors, assemble it into the other primitive cell

17 Primitive Unit cell A region of space (area or volume) that when translated through all the lattice vectors in a Bravais lattice (net) fills the entire space Without leaving voids, or overlapping itself is called a primitive unit cell

18 Properties primitive unit cell: The general choice of a primitive cell is the set of Points with position r such: r = xa 1 1+ x2a2+ x3a3 For 0 x i 1 and a, a, a Are the primitive lattice vectors. This is a parallelepiped that often times does not resemble the symmetry of the lattice at all!

19 Properties primitive unit cell: The primitive unit cell for the face centered cubic lattice. It has only ¼ of the volume of the large cube Notice that they do not have the same symmetry. Why do we say this?

20 Properties primitive unit cell: The primitive unit cell for the body centered cubic lattice. It has ½ of the volume of the large cube (conventional cell)

21 The Wigner-Seitz Primitive cell One particular primitive cell with the full symmetry of the lattice When translated through lattice Vectors it fills up all the space cell without leaving voids or overlapping with itself. The Wigner Seitz cell about one point can be constructed by: 1) Drawing lines connecting one point to all others 2) Drawing all planes(lines) bisecting each line 3) The smallest polyhedron (polygon) that encloses the point bounded by the planes (lines).

22 The Wigner-Seitz Primitive cell for the body centered cubic Bravais lattice Conventional cell Full lattice symmetry?

23 The Wigner-Seitz Primitive cell for the face centered cubic Bravais lattice This point is on the center NOT the conventional cubic cell!

24 Crystal Structure: Adding a basis to a lattice The honeycomb revisited What are the primitive vectors? The same physical unit, here two points the base is located at each point of the lattice (in the center of each parallelogram)

25 Some important examples of lattices with a basis and crystal structures The Diamond structure Center of a tetrahedron 0

26 Hexagonal Close-packed structure

27 Hexagonal Close-packed structure Hexagonal Bravais net

28 Hexagonal Close-packed structure Two Hexagonal Bravais lattices interpenetrated and displaced by c/2

29 Close-packed structures, and possibilities

30 Close-packed structures, hcp (AB)n

31 Close-packed structures, fcc (ABC)n

32 Close-packed structures, fcc (ABC)n

33 Close-packed structures, cavities

34 Sodium Chrloride structure

35 Cesiuim Chrloride structure